Imperfect not implies hypoabelian

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., imperfect group) need not satisfy the second group property (i.e., hypoabelian group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about imperfect group|Get more facts about hypoabelian group
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., imperfect group) need not satisfy the second group property (i.e., solvable group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about imperfect group|Get more facts about solvable group

Statement

It is possible for a group to be an imperfect group but not a hypoabelian group. In particular, it is possible for an imperfect group to not be a solvable group.

Proof

Finite example

For a finite group, being hypoabelian is equivalent to being a finite solvable group, so it suffices to construct a finite imperfect group that is not solvable.

The smallest example is symmetric group:S5: